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A sensitivity analysis for linear systems involving M-matrices and its application to the Leontief model. (English) Zbl 0556.15003
Let $$M^ 0$$ and $$M^*$$ be (n$$\times n)$$ non-singular M-matrices (n$$\geq 2)$$. (That is: $$M^ 0$$ and $$M^*$$ may be written as $$M^ 0=\lambda I- A^ 0$$, $$M^*=\lambda I-A^*$$ where $$\lambda$$ is a positive scalar, and $$A^ 0$$ and $$A^*$$ are non-negative matrices, and $$(M^ 0)^{- 1}$$, $$(M^*)^{-1}$$ are also non-negative.) $$M^ 0$$ and $$M^*$$ may differ only in the first s $$(0<s<n)$$ columns. Let $$w^ 0$$ and $$w^*$$ be strictly positive n-vectors, which coincide in their last (n-s) entries. Let $$p^ 0=(M^ 0)^{-1}w^ 0$$, $$p^*=(M^*)^{-1}w^*$$. Then if $$(p^ 0M^*)_ i>w^*_ i$$ for $$i\in S$$, the authors prove $$\min_{i\in S}\{p^*_ i/p^ 0_ i\}<\min_{i\in R}\{p^*_ i/p^ 0_ i\}$$, where $$S=\{1,2,...,s\}$$, $$R=\{s+1,...,n\}$$. This is a partial generalization of Theorem 21 of G. Sierksma [Linear Algebra Appl. 26, 175-201 (1979; Zbl 0409.90027)]; see also the reviewer’s paper [Non-negative matrices and Markov chains (1981; Zbl 0471.60001 pp. 35- 39], in that changes in the M-matrix $$\{$$ from $$M^ 0$$ to $$M^*\}$$ are also permitted.

##### MSC:
 15A06 Linear equations (linear algebraic aspects) 93B35 Sensitivity (robustness) 15B48 Positive matrices and their generalizations; cones of matrices 91B60 Trade models
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##### References:
 [1] Berman, A.; Plemmons, R.J., Nonnegative matrices in the mathematical sciences, (1979), Academic New York · Zbl 0484.15016 [2] Debreu, G.; Herstein, I.N., Nonnegative square matrices, Econometrica, 21, 567-607, (1953) · Zbl 0051.00901 [3] Elsner, L.; Johnson, C.R.; Neumann, M., On the effect of the perturbation of a nonnegative matrix on its Perron eigenvector, Czechoslovak math. J., 32, 99-109, (1982) · Zbl 0501.15010 [4] Fiedler, M.; Pták, V., On matrices with non-positive off-diagonal elements and positive principal minors, Czechoslovak math. J., 12, 382-400, (1962) · Zbl 0131.24806 [5] Fujimoto, T., An elementary proof of Okishio’s theorem for models with fixed capital and heterogeneous labour, Metroeconomica, 33, 21-27, (1981) [6] T. Fujimoto, C. Herrero, and A. Villar, Technical changes and their effects on the price structure, Metroeconomica, to appear. [7] Metzler, L.A., A multiple-country theory of incomes transfers, J. political economy, 59, 14-29, (1951) [8] Metzler, L.A., Taxes and subsidies in Leontief’s input-output model, Quart. J. econom., 65, 433-438, (1951) [9] Morishima, M., Equilibrium, stability and growth, (1964), Clarendon Oxford · Zbl 0117.15406 [10] Murata, Y., Mathematics for stability and optimization of economic systems, (1977), Academic New York [11] Seneta, E., Nonnegative matrices, (1973), Wiley New York · Zbl 0278.15011 [12] Seneta, E., Nonnegative matrices and Markov chains, (1981), Springer Berlin · Zbl 0471.60001 [13] Sierksma, G., Non-negative matrices: the open Leontief model, Linear algebra appl., 26, 175-201, (1979) · Zbl 0409.90027
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